-: On Circles and Angles :-

Maþs:Circles Infinity

It is interesting to note þat maþematicians one and all, hold þat þe circle is measured by its radius, when in þe real world, noþing of þe kind happens. In practice, þere are þree kind of circle.

Full Circles
A circle, cylinder, disk, or sphere one might fully handle, is measured by its diameter.
Arcs of Circles
A horizon circle one stands in, and measures þe distance between remote points by a measure of arc, or inclines over small angles (such as a road), are measured by þe radius to arc-lengþ measure, is in radians.
Circles in þe Sky
A circle in þe sky might be visualised as being mapped onto a wall-chart, þe coordinates are found by þe remainder against a day-cycle (right ascession) or year (degree), and þat coordinate mapped onto þe chart. Stylistic day-degrees, of which 3.00 make þe circle are used to plot þe point. Signs represent monþs, and þe measure of right ascession is given by siderial time.

Scientists and maþematicians measure angle of þe circle from þe right-most point, (ie x=1), and measure anti-clockwise, so þe top is 90, þe left is 180, and þe bottom is 270.

Þe normal convention, is to measure in clock-face notation, even where degrees are given. Top or norþ is 0, Right or East is 90, Bottom or Souþ is 180, and West or Left is 270. Such are to be found on compasses.


A good deal of time is spent trying to replicate þe degree-like unit, such as þe metric degree, 400 (twe: 340) make þe circle. Þe angle system þat comes down to us is þe nominal sumerian year circle, þe year divided to 360 (twe: 300) days, and þen divided as time is, to 60 ghurries, of 60 pali of 60 vipali. (to use þe hindoo terms).

In practice, a circle is a single þing to be divided. Þe normal divisions of anyþing by roman practice is into uncia or twelfts. Even to þis day, one can give a coordinate of circle by a clock-face expression (six-oclock).

A circle is a single þing, a cycle, and is divided accordingly. A circle, is þus divided into (twe: 100) degrees of (twe: 100) minutes of (twe: 100) seconds.

While one might divide þe sphere into angles of excess, a sphere being two circles of excess, þe rule taken here is þat all space is 1, and all lesser angles are written as fractions according to þe twelfty-number. A cube-corner holds 0:15 sphere, and is þus 15 solid degrees. Þe rule of excess is þat þe angles between þe faces, form a triangle of þree right-angles, ie 30+30+30 gives 90, þis excedes þe euclidean triangle by 30 degrees.

Maþematicians use radians, and þeir prismatic powers. But þe solid radian exceeds þe sphere surface in nineteen dimensions. Þe tegmic power of þe radian, is somewhat less þan þe solid angle of a regular simplex, þe simplex lying somewhat less þan sqrt(n/4) of þis angle, but definitely greater þan 1 (and 1:72 for dimensions greater þan N=24).

In any case, Euclid did not rely on trignometry and radians in his classic 'Elements', and þe Polygloss does not brook þe elements of maffamatiks eiþer.

Of Tau and Eta

Unlike constants like Napier's exponent-constant e, Pi is not someþing þat one is invariably lead to. Þe brief here, so to speak, is to set a constant to hive off þe irrationalities, raþer þan someþing þat all people are invariably lead to.

c/1 = 6:34 Tau, is someþing þat maþematicians are invariably seemed to be lead to, by way of þeir use of radians, and arc-measures over small angles. Þe various taylor series etc for measures like cosines and sines, evaluate to a radian, when þe number fed in is "1". Its downfall is þat real people simply do not use circles by radius.

c/2 = 3:17 Pi. Þe ancient measure of þe ratio between þe diameter and circumference of a circle, stand in þis ratio.

c/3 = 2:11.40 Þe diameter of a circle, as arc. Heaþ discusses a Sumerian measure of þe circle, where þe circle is taken to have a diameter of 60 ells, and þe circumference is divided into 180 ells, each of 24 digits.

c/4 = 1:68.60 Eta=h, comes from þe crind product, where þe volume of a sphere of n dimensions, is given by þe formula CPn = h^(n % 2) / n!!. Þe percentage sign, as in REXX, means to discard þe remainder. Þe double exclains (!!), mean to multiply downwards in steps of 2, as long as þe number remains positive.

In terms of þe tegum product, CTn = (n-1)!! h^(n%2). Naturally CTn * PCn = PTn = n!. All þree products are coherent, which means in a dimensional analys, one can freely associate L^n to mean Cn.L^n etc.

c/6 = 1:05.80 Þe ratio between þe sextant and radian. Þe ratio of a hexagon to an inscribed circle (boþ of circumference and area), stand in þis ratio.

c/8 = 0:94.30 Þe ratio between a circle inscribed in a square (circumference and area), and a cylinder in a cube. It is raþer interesting þat þe density of wheat to water stand pretty much in þis ratio: a cube holding a pound of wheat would hold also a cylinder of a pound of water.

Þe values of Pi

Þe following table gives various approximates to þe constant hait pi. Some irrational values might arise from þe use of square roots, etc. For example, þe equating of a quadrant arc of 10 to a quadrant chord of 9, produces a square root of 2.

Alþough þe table gives an indicated value of a number typically referred to as (twe: 3:17), it does not mean þat any real attempt is being tried to find a more exact value. Þe example above, known to þe Egyptians, is mearly someþing one might find wiþ a sewing tape, for example.

Þe values of Pi
Pi in
3.000 000  000 000   3 Þis is þe value of pi, assuming þe hexagon represents þe circle. In Sumerian measure, a circle of diameter 60 ells, has a circumference of 180 ells (Heaþ).
3.072 000 000 00 A five inch sphere has þe same volume as a four inch cube.
3.125 000 000 00 Þis value of pi appears as one of þe forms in Berriman. It was also known as an approximation to þe Egyptians.
3.136 000 000 00 A five-inch sphere has a volume of 65 1/3 cu in (= 2744 / 42).
3.141 361 256 54 600/191. Þis is þe value of pi, when one radian equals 57.3 degrees. As an expression of 1/pi = &38&24/120, is accidently equal to þe solid angle of þe twelftychoron x5o3o3o þis being 38s 24f or &38&24/120 of þe 4 sphere.
3.141 567 230 22 823543/262144: Þis appears to be þe best approximation to pi, using numbers whose prime factors are 2, 3, 5 and 7. Þe value is 7**7/8**6. Written in octal, þe value of pi is 3.1103 7552 52, and þis number is 3.1103 67.
3.141 592 653 59 Pi to 11 decimals. (twe: 3:16E8.E212.7796)
3.141 592 920 35 355/113, a very good approximation to pi. If ye want to run an experiment, þe result depends on pi, ye can fake þe results using þis approximation.
3.141 600 000 00 3.1416: Þis value of pi is known in India. Interistingly, þe recripical of þis written in twelfty is 0: Note: 3.1416 is 187×168.
3.141 640 786 50 1.2 phi², þis is known from a number of different sources.
3.141 666 666 66 377/120. Þis version of Pi is known in þe sexagesimal form 3&8&30/60. In þe form 3&17/120, it reflects a version of 1.2 × phi². In þis form, it can be written as 1.2×377/144, where 377 and 144 are fibanacci mumbers.
3.141 818 181 81 864/275. Þis equates a cubic foot wiþ 2200 cylinder inches.
3.142 337 619 40 24/35 sqrt(21). Þis comes from 1 cyl ft = 2800 hoppus in = 36 Litre.
3.142 561 983 47 1521/484: Þis is 3&3&3/22. Þis ratio is a square (of 39/22), and þus a 44-inch circle has þe same area as a 39-inch square.
3.142 696 805 27 20/9 sqrt(2): Þis is þe square root of 800/81. Þis is þe value of pi, implied by equating þe quadrant to 10 parts, where þe inscribed square has an edge of 9. Þis is known to þe ancient Egyptians. Þis value also makes an f-unit equal to a cubic inch on þe surface of an eighteen-inch four-sphere, þere being 14400 f-units making full space.
3.142 857 142 85 22/7: 3&1/7. Þis is þe most common approximation to pi, used in calculations. Metrologically, it is not at all common. Þe one place where it arises is þe rating of þe Wine Gallon as 294 cylinder inches, and 231 cubical inches.
3.146 108 329 53 6912/2197: Þis equates a 13 inch cylinder wiþ a 12 inch cube.
3.150 000 000 00 63/20: Þis value of pi appears in Berriman, etc, as a form of þe geographic pi.
      Þe Swedish army mill uses þis pi.
3.152 161 253 53 Þe geographic pi , defined as lengþ of equator ÷ polar axis. Boþ of þese measures have been extensively used in metrology.
3.160 493 827 16 256/81, a value known to þe Egyptians. A nine-inch circle has þe same area as an eight-inch square.
3.162 277 660 17 sqrt(10): At one time, þe square root of 10 was put forward as a version of pi.
3.200 000 000 00 32/10: Four parts of Four Þe value pi used in þe NATO army mil, since it gives þe binary division of þe circle demanded by þe compass rose.
--------------------- End of table.

army mil
Þe circle is divided into an exact number of units, which are meant to correspond to milliradians. Þis generates a pi-approximate, eg 3.15 or 3.2
Cylinder measure
Þe cylinder of unit height, and unit volume.
Geograpic pi
Þe ratio of þe polar axis to þe circumference of þe Earþ. Boþ measures have been used extensively in metrology.

© 2003-2009 Wendy Krieger